9709 Pure Mathematics 1 Oct/Nov 2023 Paper 13

24 questions and parts, 75 marks in total. Marks come from the published mark scheme.

1. Question 1

   4 marks · 4/75

   This assesses integration of a given gradient followed by using a point on the curve to determine the constant of integration.

   Examiner report: This was a strong opening question for most candidates, who usually integrated successfully and used the given point correctly.

   Question paper p.3 · Mark scheme p.6

2. Question 2 (a)

   2 marks · 6/75

   This assesses finding the intersections of a circle with the y-axis and expressing the resulting coordinates using surds.

   Examiner report: Most candidates had little difficulty imposing the y-axis condition and solving the resulting equation; completing the square was also an effective route.

   Question paper p.4 · Mark scheme p.6

3. Question 2 (b)

   2 marks · 8/75

   This assesses constructing the equation of a circle from the endpoints of its diameter.

   Examiner report: The centre and radius were usually identified correctly, though some candidates used the full diameter as the radius or misplaced the centre horizontally.

   Question paper p.4 · Mark scheme p.6

4. Question 3 (a)

   3 marks · 11/75

   This assesses transforming a trigonometric equation containing tan θ into a quadratic in cos θ using identities.

   Examiner report: Most candidates selected the necessary identities and handled the algebra effectively, with only occasional errors caused by brackets or fraction removal.

   Question paper p.5 · Mark scheme p.7

5. Question 3 (b)

   4 marks · 15/75

   This assesses solving a trigonometric equation by factoring a quadratic in cos θ and finding every angle in a specified interval.

   Examiner report: Although many solutions were complete, some candidates relied on calculator root-solving without displaying the factorisation, and others omitted solutions from one of the relevant quadrants.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (a)(i)

   1 mark · 16/75

   This assesses using the binomial theorem to expand a positive integer power through the quadratic term.

   Examiner report: This expansion was handled correctly in almost all scripts, with a few candidates either mishandling powers or writing unnecessary later terms.

   Question paper p.6 · Mark scheme p.7

7. Question 4 (a)(ii)

   2 marks · 18/75

   This assesses binomial expansion when the linear term contains a negative constant and the result is required only through degree two.

   Examiner report: Most candidates applied the theorem well; the main slips involved the sign of the linear contribution and the powers attached to a.

   Question paper p.6 · Mark scheme p.7

8. Question 4 (b)

   4 marks · 22/75

   This assesses extracting a coefficient from the product of two truncated binomial expansions and solving the resulting quadratic in a.

   Examiner report: The coefficient calculation was often sound, but some candidates lost a mark by presenting the quadratic roots without showing an algebraic solution.

   Question paper p.6 · Mark scheme p.8

9. Question 5 (a)

   3 marks · 25/75

   This assesses using the common-ratio property of a geometric progression to determine an unknown constant.

   Examiner report: Most candidates established the ratio relationship successfully and found the two values, while those who introduced a cubic generally found the later algebra more difficult.

   Question paper p.7 · Mark scheme p.8

10. Question 5 (b)

    4 marks · 29/75

    This assesses finding the sum to infinity of a geometric progression after selecting a specified value of its parameter.

    Examiner report: The sum formula was widely known, but some candidates treated the chosen parameter itself as the common ratio instead of calculating the ratio from consecutive terms.

    Question paper p.7 · Mark scheme p.8

11. Question 6

    7 marks · 36/75

    This assesses the condition for a line and a quadratic curve to be tangent, together with finding the associated contact points.

    Examiner report: Both the discriminant method and the equal-gradient method were popular and often successful; a common omission was failing to show a genuine solution of the quadratic involved.

    Question paper p.8 · Mark scheme p.9

12. Question 7 (a)

    1 mark · 37/75

    This assesses determining the range of a restricted rational function.

    Examiner report: This was generally answered well, although some responses described an input interval rather than the requested set of output values.

    Question paper p.9 · Mark scheme p.10

13. Question 7 (b)

    4 marks · 41/75

    This assesses finding the inverse of a one-to-one rational function and identifying its domain from the original range.

    Examiner report: The inverse procedure was familiar to most candidates; lost marks mainly came from sign errors or from giving the domain in the wrong variable.

    Question paper p.9 · Mark scheme p.10

14. Question 7 (c)

    2 marks · 43/75

    This assesses composing two functions in the stated order and simplifying the resulting rational expression.

    Examiner report: This was the most consistently successful part, with the order of composition and the simplification usually made clear.

    Question paper p.9 · Mark scheme p.11

15. Question 8 (a)

    2 marks · 45/75

    Question paper p.10 · Mark scheme p.11

16. Question 8 (b)

    3 marks · 48/75

    This assesses tracking the coordinates of a stationary point under horizontal scaling and translation together with vertical reflection and stretch.

    Examiner report: Many candidates understood the vertical change, but fewer correctly combined the horizontal scale with the translation; a frequent error was to apply the shift without the scale factor.

    Question paper p.10 · Mark scheme p.11

17. Question 9 (a)

    3 marks · 51/75

    This assesses differentiating a curve, obtaining the normal gradient at a specified point, and forming its straight-line equation.

    Examiner report: Most candidates differentiated and found the normal successfully, with the main work being the correct conversion from tangent gradient to normal gradient.

    Question paper p.11 · Mark scheme p.12

18. Question 9 (b)

    2 marks · 53/75

    This assesses applying the chain rule to relate the rates of change of the coordinates of a moving point.

    Examiner report: The chain rule was commonly used correctly, though some candidates left the gradient in terms of x instead of evaluating it at the specified point.

    Question paper p.11 · Mark scheme p.12

19. Question 9 (c)

    3 marks · 56/75

    This assesses using a known normal gradient and the chain rule to find the horizontal velocity when the vertical velocity is specified.

    Examiner report: Most candidates recognised the need for the chain rule, but the normal gradient from the earlier part and the sign associated with decreasing y were not always carried through.

    Question paper p.11 · Mark scheme p.12

20. Question 10 (a)

    1 mark · 57/75

    Question paper p.12 · Mark scheme p.13

21. Question 10 (b)

    1 mark · 58/75

    Question paper p.12 · Mark scheme p.13

22. Question 10 (c)

    7 marks · 65/75

    This assesses finding a composite area bounded by circular arcs by combining sector and triangle or segment areas.

    Examiner report: A range of valid decompositions was seen, but many solutions were difficult to follow because intermediate geometry and area calculations were not clearly labelled or explained.

    Question paper p.13 · Mark scheme p.13

23. Question 11 (a)

    4 marks · 69/75

    This assesses differentiating a rational-power expression, verifying a stationary point at a specified x-coordinate, and finding its ordinate.

    Examiner report: Differentiation was usually accurate, but the verification step mattered: candidates needed to show explicitly that the given coordinate makes the derivative vanish.

    Question paper p.14 · Mark scheme p.14

24. Question 11 (b)

    6 marks · 75/75

    This assesses calculating the area between a curve and the chord joining its two specified intersection points.

    Examiner report: Most candidates identified the two required regions and integrated the curve competently; errors arose from combining integrals carelessly or failing to show how the limits were applied.

    Question paper p.15 · Mark scheme p.15